Number System Practice Gym by Anubhav Sehgal  Part 2

Q1) Find the sum of all possible values of x such that 2/(sqrt(x)) + 1/(sqrt(y)) = 1/(sqrt(2)) where x and y are positive integers
u = sqrt(x) and v = sqrt(y)
2v + u = uv/rt(2)
uv  2rt2 v  rt2 u = 0
v(u  2rt2)  rt2(u  2rt2) = 4
(v  rt2)(u  2rt2) = 4 = rt2 * 2rt2 = 2rt2 * rt2
u,v = 4rt2,2rt2 ; 3rt2,3rt2
x,y = 32,8 ; 18,18
Sum = 50Q2) How many three digit numbers are 34 times of sum of their sum of digits?
100a + 10b + c = 34a + 34b + 34c
66a  24b  33c = 0
66a = 24b + 33c
a = 1 ; b = 0 ; c = 2
a = 2 ; b = 0 ; c = 4
a = 3 ; b = 0 ; c = 6
a = 4 ; b = 0 ; c = 8
Four such numbers.Q3) Find the number of integers, N such that (N^2 + 2N  8 )/(N^2 + N  12) is an integer.
N^2 + 2N  8 = (N  2)(N + 4)
N^2 + N  12 = (N  3)(N + 4)
=> (N  2)/(N  3) is an integer
=> (N  3 + 1)/(N  3) is an integer
=> 1 + 1/(N  3) is an integer => N = 4. Only one possible value.Q4) Find the 1000th term of the sequence : 1,3,4,7,8,9,10,11,13,14,... in which there is no number which contain digit 2,5 or 6.
I have two approaches for this one :
The Shareef approach
The Tareef approachThe Shareef approach :
_ : 6 numbers ; _ _ : 6 * 7 = 42 numbers ; _ _ _ : 6 * 7 * 7 = 294 numbers
342 numbers added till 999
1 _ _ _ : 7 * 7 * 7 = 343 added = 685 numbers till now. Remaining 1000  685 = 315
3 0 _ _ : 49 added
3 1 _ _ : 49 added
3 3 _ _ : 49 added
3 4 _ _ , 3 7 _ _ , 3 8 _ _ = 49*3 = 147
Total 294 more. Remaining 21
3 9 0 _ : 7
3 9 1 _ : 7
3 9 3 _ : 7
3939The Tareef approach :
0 > 0
1 > 1
2 > 3
3 > 4
4 > 7
5 > 8
6 > 9
1000th term : 1000 in base 7 = 2626 3939Q5) How many positive divisors of 100^10 end in exactly two zeroes?
100^10 = 2^20 * 5^20
2^18 * 5^18
19 + 19  1 = 37Q6) 960 = 880
c + 880 + c + 800 + c = 1680 + 3c > 1000 Not to be considered
OR
for x c = 1 to 13
x = 21 to 8
Option c)Q7) How many pairs of non negative integers exist,such that the difference between their product and sum is 72?
ab  a  b = 72
ab  a  b + 1 = 73
a(b  1)  1(b  1) = 73
(a  1)(b  1) = 73 = 1 * 73 = 1 * 73
a,b = 2,74 ; 74,2 ; 0,72 ; 72,0
Non negative integer pairs = 1 unordered or 2 ordered
Total integral solutions = 2 unordered or 4 ordered
Option d) Answer : 1Q8) The equation 3^(x  1) + 5^(x  1) = 34 has :
a) No solution
b) 1 solution
c) 2 solutions
d) More than 2 solutions34 = 25 + 9
x = 3 is the only solution
Both are exponentially increasing.
3 + 5 = 8< 34 ; 9 + 25 = 34 ; 27 + 125 > 34. No other values. Option b)Q9) For what smallest positive integer N, will the number 3! + 5! + 7! + .. + (2N + 1)! be a perfect square ?
3!(1 + 4 * 5 + 4 * 5 * 6 * 7 + ....) = 3! * odd . Power of 2 is 1. So no possible N.
Q10) What is the unit digit of [10^3000/(10^100 + 3)] where [.] is GIF.
Remainder = 10^3000/(10^100 + 3)
Put x = 10^100 => x^30/x + 3. Put x = 3 => (3)^30 = 3^30
Dividend = Quotient*Divisor + Remainder
Check for unit digit on both sides,
0 = x * 3 + 9
x = 7